Digital Library of Mathematical Functions
The Gamma Function -- Sums

Sums

∑ k = 1 ∞ ( - 1 ) k ψ ′ ( k ) = - π 2 8
∑ k = 1 ∞ 1 k ψ ′ ( k + 1 ) = ζ ( 3 ) = - 1 2 ψ ′ ′ ( 1 )

For further sums involving the psi function see Hansen(1975) (pp. 360–367). For sums of gamma functions see Andrews et.al.(1999) (Chapters 2 and 3).

For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et.al.(1999) (Chapter 1) and Terras(1999) .